Posing armature causes mesh to stretch
It looks like some vertices are getting left without any vertex weight influence. While in Edit Mode, select one single instance of any of these straggling vertices and check the Properties Shelf (to the right of the 3D View) for the Item Tab, which brings up the Transform Panel. Then look to the area beneath the Transform Panel for the Vertex Weights section. For any vertex that is selected, you will be shown all the vertex groups that the currently selected vertex is currently part of, as well as the vertex weighting assigned to each of the listed vertex groups. If you don't see any Vertex Weight section at all, it means the selected vertex is not part of any vertex group, and that would be your problem. Assign them by navigating to the Object Data tab in the Properties Panel, then look for the Vertex Groups Section, where you can assign, remove, and adjust the influence (weight) of the currently selected vertices.